proceedings of the american mathematical

Volume 28, No. 1, April 1971

society

PARACOMPACTNESS AND ELASTIC SPACES HISAHIRO TAMAÑO1 AND J. E. VAUGHAN Abstract. This paper gives a characterization of paracompactness, and introduces the notion of an elastic space which generalizes the concept of a stratifiable (in particular, metric) space.

1. Introduction. In this note we shall give a characterization of paracompactness which is formally weaker than our previous characterizations [4, Theorem 2], [5, Theorem 3], [6, Theorem l] concerning linearly cushioned refinements. Furthermore, we shall define a new generalization of metric spaces and stratifiable spaces, called "elastic spaces," by introducing the notion of an "elastic base." Definition 1. Let It be a collection of subsets of a set X, and let

(ft be a relation on It (i.e., (RCltXIt). instead

of ( U, V) E 0"LThe relation

We shall often write UGlV

(ft is said to be a framed

relation

on 11 (or a framing of It) provided for every U, FGlf, if Ur\V?i0, then USi V or F (ft U. We say (ft is a well-framed relation on It provided (ft is a framing of 11 and for every xEX, there exists an (ft-smallest

c7IGcUcontainingx(i.e.,ifxGZ7, Definition 2. A collection

f/Gll, and U¿¿ Ux,then iU, UX)E(R). 11 is said to be framed in a collection V with frame map f: 11—»*U. provided there exists a framed relation (ft on 11 such that for every subcollection It'CU which has an (ft-upper bound (i.e., there exists £/Glt so that U'GlU for every C/'Glt') we

have cl(Ull')CU/(1t').

If in addition

(ft is a well-framed relation on

It, we say that It is well-framed in V. Finally, if It is framed in V and (ft is also a transitive relation, then 11 is called elastic in V, or an elastic refinement of V when It and V are covers of X.

Theorem 1. Let X be a regular space. A necessary and sufficient condition that X be paracompact is that every open cover of X have an open elastic refinement.

2. Proof lemmas. Presented

of Theorem

1. The proof follows from the next two

to the Society, August 28, 1970; received by the editors September

22,

1969. AMS 1969 subject classifications. Primary 5450, 5435. Key words and phrases. Paracompact, framing, elastic refinement, elastic base, elastic space, Jkf3-space, stratifiable space, metric space. 1 This note is a revision of a manuscript of H. Tamaño which was found by J. Nagata after the death of Professor Tamaño. Copyright

299

© 1971, American

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Mathematical

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hisahiro tamaño and j. e. vaughan

300

[April

Lemma 1. Let X be a regular space. A necessary and sufficient condition for X to be paracompact is for every open cover of X to have an open refinement which is well-framed in it. Proof. We shall prove the sufficiency. The proof is similar to that of Theorem 1 in [ó]. Let V be an open cover of X. Let 11 be an open refinement of V which is well-framed in V with respect to the wellframed relation 01 on 11 and frame map /¡It—>U. Let HV=U —

\J { U'E^: U'&U and U'^U}, and 3C= {Hv: Z/Gll}. We now show that X is a cushioned refinement of V with cushion defined by g(Hv) =f(U) (these terms are defined in [ô]) that X is paracompact by [3, Theorem 1.1, p. 309]. It that 3C is a cover of X since (R is well-framed. It remains

map g:3C—>V and conclude is easy to see to show that

3C is cushioned in TJ. Let 3C'C3C, and suppose x^Ug(X').

Let Ux be

an (R-smallest element of 11 containing x. Clearly Ux is an open neighborhood of x missing Hxj for all U^UX such that Ux(ñU. Further,

11'= {t/Gli: U*y. If a